Advanced/Retarded Eddington-Finkelstein Coordinates and Incident/Outgoing Wave

[yicong2011]

My professor said that

Advanced Eddington-Finkelstein Coordinates CAN be applied to describe incident wave well, while CANNOT describe outgoing wave.

Conversely, Retarded Eddington-Finkelstein Coordinates CAN be applied to descibe outgoing wave well, while CANNOT describe incident wave well.

I cannnot understand why...

Anyone can help to explain it?

Thanks...


(Neither y professor nor I are native English speakers. Perhaps language jeopardises the understanding..)

 

[Bill_K]

Schwarzschild coordinates only cover part of the spacetime (outside r=2m). Both outgoing and incoming waves run off the coordinate patch.

Eddington coordinates cover more of the spacetime, but still not all of it. As you say, advanced coordinates work well for incoming waves, but outgoing waves run off the patch. Likewise for retarded coordinates, outgoing waves are ok but incoming waves run off the patch.

Kruskal coordinates cover the entire spacetime and can be used for both.

 

[JesseM]

Eddington coordinates cover more of the spacetime, but still not all of it. As you say, advanced coordinates work well for incoming waves, but outgoing waves run off the patch. Likewise for retarded coordinates, outgoing waves are ok but incoming waves run off the patch.

Yes, as discussed here the "maximally extended" version of the spacetime contains both an interior black hole region and a separate interior white hole region, and advanced coordinates contain only the interior black hole region while retarded coordinates containe only the interior white hole region (and neither contains the other exterior region, region III in the diagram in the link). Another aspect of the problem is that in retarded coordinates, particles leaving from the white hole region cross the event horizon into the exterior region at some finite coordinate time, but particles falling in from the event horizon never actually reach it at any finite coordinate time, instead taking an infinite coordinate time to get arbitrarily close to it. In contrast, in advanced coordinates the opposite is true--particles falling in due reach the event horizon at a finite coordinate time, but particles leaving the horizon have been moving away from it for an infinite coordinate time in the past.

From this page, a diagram of advanced coordinates showing world lines both of light rays falling into the horizon and also coming out:

[URL]http://ckw.phys.ncku.edu.tw/public/pub/Notes/TheoreticalPhysics/Lawrie_2/Chap04/4.5.4._EddingtonFinkelsteinCoordinates.files/image001.jpg[/URL]

You can see the ones coming out never actually crossed the event horizon at any point on the diagram, and even if you extended the diagram back to earlier times these light rays would always be slightly outside the event horizon at any time in the past. On the other hand, here's the diagram for retarded coordinates:

[URL]http://ckw.phys.ncku.edu.tw/public/pub/Notes/TheoreticalPhysics/Lawrie_2/Chap04/4.5.4._EddingtonFinkelsteinCoordinates.files/image002.jpg[/URL]

Here you can see light rays crossing the horizon in the outward direction at finite times, but the ones falling in never quite reach it, though they get closer and closer.